
Physica D: Nonlinear Phenomena [SJR: 1.048] [HI: 89] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 01672789 Published by Elsevier [2970 journals] 
 Fractional Schrödinger dynamics and decoherence
 Abstract: Publication date: Available online 16 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Kay Kirkpatrick, Yanzhi Zhang
We study the dynamics of the Schrödinger equation with a fractional Laplacian ( − Δ ) α , and the decoherence of the solution is observed. Analytically, we obtain equations of motion for the expected position and momentum in the fractional Schödinger equation, equations that are the fractional counterpart of the wellknown Newtonian equations of motion for the standard ( α = 1 ) Schrödinger equation. Numerically, we propose an explicit, effective numerical method for solving the timedependent fractional nonlinear Schrödinger equation–a method that has high order spatial accuracy, requires little memory, and has low computational cost. We apply our method to study the dynamics of fractional Schrödinger equation and find that the nonlocal interactions from the fractional Laplacian introduce decoherence into the solution. The local nonlinear interactions can however reduce or delay the emergence of decoherence. Moreover, we find that the solution of the standard NLS behaves more like a particle, but the solution of the fractional NLS behaves more like a wave with interference effects.
PubDate: 20160618T18:04:02Z
 Abstract: Publication date: Available online 16 June 2016
 Entropy rates of lowsignificance bits sampled from chaotic physical
systems Abstract: Publication date: Available online 16 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Ned J. Corron, Roy M. Cooper, Jonathan N. Blakely
We examine the entropy of lowsignificance bits in analogtodigital measurements of chaotic dynamical systems. We find the partition of measurement space corresponding to lowsignificance bits has a corrugated structure. Using simulated measurements of a map and experimental data from a circuit, we identify two consequences of this corrugated partition. First, entropy rates for sequences of lowsignificance bits more closely approach the metric entropy of the chaotic system, because the corrugated partition better approximates a generating partition. Second, accurate estimation of the entropy rate using lowsignificance bits requires long block lengths as the corrugated partition introduces more longterm correlation, and using only short block lengths overestimates the entropy rate. This second phenomenon may explain recent reports of experimental systems producing binary sequences that pass statistical tests of randomness at rates that may be significantly beyond the metric entropy rate of the physical source.
PubDate: 20160618T18:04:02Z
 Abstract: Publication date: Available online 16 June 2016
 Nonlinear wave dynamics near phase transition in PTsymmetric localized
potentials Abstract: Publication date: 15 September 2016
Source:Physica D: Nonlinear Phenomena, Volume 331
Author(s): Sean Nixon, Jianke Yang
Nonlinear wave propagation in paritytime symmetric localized potentials is investigated analytically near a phasetransition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for a phase transition to occur are derived based on a generalization of the Krein signature. Using the multiscale perturbation analysis, a reduced nonlinear ordinary differential equation (ODE) is derived for the amplitude of localized solutions near phase transition. Above the phase transition, this ODE predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodicallyoscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below the phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.
PubDate: 20160618T18:04:02Z
 Abstract: Publication date: 15 September 2016
 Degenerate Bogdanov–Takens bifurcations in a onedimensional
transport model of a fusion plasma Abstract: Publication date: 15 September 2016
Source:Physica D: Nonlinear Phenomena, Volume 331
Author(s): H.J. de Blank, Yu.A. Kuznetsov, M.J. Pekkér, D.W.M. Veldman
Experiments in tokamaks (nuclear fusion reactors) have shown two modes of operation: Lmode and Hmode. Transitions between these two modes have been observed in three types: sharp, smooth and oscillatory. The same modes of operation and transitions between them have been observed in simplified transport models of the fusion plasma in one spatial dimension. We study the dynamics in such a onedimensional transport model by numerical continuation techniques. To this end the MATLAB package cl_matcontL was extended with the continuation of (codimension2) Bogdanov–Takens bifurcations in three parameters using subspace reduction techniques. During the continuation of (codimension2) Bogdanov–Takens bifurcations in 3 parameters, generically degenerate Bogdanov–Takens bifurcations of codimension3 are detected. However, when these techniques are applied to the transport model, we detect a degenerate Bogdanov–Takens bifurcation of codimension 4. The nearby 1 and 2parameter slices are in agreement with the presence of this codimension4 degenerate Bogdanov–Takens bifurcation, and all three types of L–H transitions can be recognized in these slices. The same codimension4 situation is observed under variation of the additional parameters in the model, and under some modifications of the model.
PubDate: 20160618T18:04:02Z
 Abstract: Publication date: 15 September 2016
 Burgers equation with noflux boundary conditions and its application for
complete fluid separation Abstract: Publication date: 15 September 2016
Source:Physica D: Nonlinear Phenomena, Volume 331
Author(s): Shinya Watanabe, Sohei Matsumoto, Tomohiro Higurashi, Naoki Ono
Burgers equation in a onedimensional bounded domain with noflux boundary conditions at both ends is proven to be exactly solvable. Cole–Hopf transformation converts not only the governing equation to the heat equation with an extra damping but also the nonlinear mixed boundary conditions to Dirichlet boundary conditions. The average of the solution v ¯ is conserved. Consequently, from an arbitrary initial condition, solutions converge to the equilibrium solution which is unique for the given v ¯ . The problem arises naturally as a continuum limit of a network of certain microdevices. Each microdevice imperfectly separates a target fluid component from a mixture of more than one component, and its input–output concentration relationships are modeled by a pair of quadratic maps. The solvability of the initial boundary value problem is used to demonstrate that such a network acts as an ideal macroseparator, separating out the target component almost completely. Another network is also proposed which leads to a modified Burgers equation with a nonlinear diffusion coefficient.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: 15 September 2016
 Spikeadding in parabolic bursters: The role of foldedsaddle canards
 Abstract: Publication date: Available online 31 May 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Mathieu Desroches, Martin Krupa, Serafim Rodrigues
The present work develops a new approach to studying parabolic bursting, and also proposes a novel fourdimensional canonical and polynomialbased parabolic burster. In addition to this new polynomial system, we also consider the conductancebased model of the Aplysia R15 neuron known as the Plant model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely the BaerRinzelCarillo (BRC) phase model. Revisiting these models from the perspective of slowfast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spikeadding process occurs in an explosive fashion that involves special solutions called canards. This spikeadding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across all considered systems, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a foldedsaddle singularity. However there can be a difference in the spikeadding transitions in parameter space from one case to another, according to whether the process is continuous or discontinuous, which depends upon the geometry of the foldedsaddle canard. Using these findings, we construct a new polynomial approximation of the Plant model, which retains all the key elements for parabolic bursting, including the spikeadding transitions mediated by foldedsaddle canards. Finally, we briefly investigate the presence of spikeadding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: Available online 31 May 2016
 Dynamics of transcriptiontranslation networks
 Abstract: Publication date: Available online 2 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): D. Hudson, R. Edwards
A theory for qualitative models of gene regulatory networks has been developed over several decades, generally considering transcription factors to regulate directly the expression of other transcription factors, without any intermediate variables. Here we explore a class of models that explicitly includes both transcription and translation, keeping track of both mRNA and protein concentrations. We mainly deal with transcription regulation functions that are steep sigmoids or step functions, as is often done in proteinonly models, though translation is governed by a linear term. We extend many aspects of the proteinonly theory to this new context, including properties of fixed points, description of trajectories by mappings between switching points, qualitative analysis via a statetransition diagram, and a result on periodic orbits for negative feedback loops. We find that while singular behaviour in switching domains is largely avoided, nonuniqueness of solutions can still occur in the stepfunction limit.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: Available online 2 June 2016
 Combustion waves in hydraulically resistant porous media in a special
parameter regime Abstract: Publication date: Available online 7 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Anna Ghazaryan, Stéphane Lafortune, Peter McLarnan
In this paper we study the stability of fronts in a reduction of a wellknown PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: Available online 7 June 2016
 MixedMode Oscillations in a piecewise linear system with multiple time
scale coupling Abstract: Publication date: Available online 10 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): S. FernándezGarcía, M. Krupa, F. Clément
In this work, we analyze a four dimensional slowfast piecewise linear system with three time scales presenting MixedMode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: Available online 10 June 2016
 A principle of similarity for nonlinear vibration absorbers
 Abstract: Publication date: Available online 11 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): G. Habib, G. Kerschen
This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: Available online 11 June 2016
 Resonance Van Hove singularities in wave kinetics
 Abstract: Publication date: Available online 14 June 2016
Source:Physica D: Nonlinear Phenomena
Author(s): YiKang Shi, Gregory L. Eyink
Wave kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. However, we show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electronhole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D = ( N − 2 ) d ( d physical space dimension, N the number of waves in resonance) and the degree of degeneracy δ of the critical points. Following Van Hove, we show that nondegenerate singularities lead to finite phase measures for D > 2 but produce divergences when D ≤ 2 and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when D − δ ≤ 2 , as we find for several physical examples, including electronhole kinetics in graphene. When the standard kinetic equation breaks down, then one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multiscale perturbation theory for acoustic waves and fieldtheoretic methods based on exact SchwingerDyson integral equations for the wave dynamics.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: Available online 14 June 2016
 Dynamics of curved fronts in systems with powerlaw memory
 Abstract: Publication date: 1 August 2016
Source:Physica D: Nonlinear Phenomena, Volumes 328–329
Author(s): M. Abu Hamed, A.A. Nepomnyashchy
The dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via twodimensional fractional in time partial differential equation. A closed equation governing a slow motion of a smallcurvature front is derived and applied for two typical examples of the potential function. Approximate axisymmetric and nonaxisymmetric solutions are obtained.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: 1 August 2016
 Breathers in a locally resonant granular chain with precompression
 Abstract: Publication date: 15 September 2016
Source:Physica D: Nonlinear Phenomena, Volume 331
Author(s): Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein
We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schrödinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, timeperiodic states mounted on top of a nonvanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.
PubDate: 20160615T08:40:59Z
 Abstract: Publication date: 15 September 2016
 Chevron folding patterns and heteroclinic orbits
 Abstract: Publication date: Available online 11 May 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Christopher J. Budd, Amine N. Chakhchoukh, Timothy J. Dodwell, Rachel Kuske
We present a model of multilayer folding in which layers with bending stiffness E I are separated by a very stiff elastic medium of elasticity k 2 and subject to a horizontal load P . By using a dynamical systems analysis of the resulting fourth order equation, we show that as the end shortening per unit length E is increased, then if k 2 is large there is a smooth transition from small amplitude sinusoidal solutions at moderate values of P to larger amplitude chevron folds, with straight limbs separated by regions of high curvature when P is large. The chevron solutions take the form of near heteroclinic connections in the phaseplane. By means of this analysis, values for P and the slope of the limbs are calculated in terms of E and k 2 .
PubDate: 20160516T19:32:17Z
 Abstract: Publication date: Available online 11 May 2016
 Stabilisation of difference equations with noisy predictionbased control
 Abstract: Publication date: 1 July 2016
Source:Physica D: Nonlinear Phenomena, Volume 326
Author(s): E. Braverman, C. Kelly, A. Rodkina
We consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to predictionbased control. These perturbations may be multiplicative x n + 1 = f ( x n ) − ( α + l ξ n + 1 ) ( f ( x n ) − x n ) , n = 0 , 1 , … , if they arise from stochastic variation of the control parameter, or additive x n + 1 = f ( x n ) − α ( f ( x n ) − x n ) + l ξ n + 1 , n = 0 , 1 , … , if they reflect the presence of systemic noise. We begin by relaxing the control parameter in the deterministic equation, and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval. Then, by allowing the variation to be stochastic, we derive sufficient conditions (less restrictive than known ones for the unperturbed equation) under which the positive equilibrium will be globally a.s. asymptotically stable: i.e. the presence of noise improves the known effectiveness of predictionbased control. Finally, we show that systemic noise has a “blurring” effect on the positive equilibrium, which can be made arbitrarily small by controlling the noise intensity. Numerical examples illustrate our results.
PubDate: 20160516T19:32:17Z
 Abstract: Publication date: 1 July 2016
 A conservation law model for bidensity suspensions on an incline
 Abstract: Publication date: Available online 11 May 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Jeffrey T. Wong, Andrea L. Bertozzi
We study bidensity suspensions of a viscous fluid on an incline. The particles migrate within the fluid due to a combination of gravityinduced settling and shear induced migration. We propose an extension a recent model (Murisic et al., 2013) for monodisperse suspensions to two species of particles, resulting in a hyperbolic system of three conservation laws for the height and particle concentrations. We analyze the Riemann problem and show that the system exhibits threeshock solutions representing distinct fronts of particles and liquid traveling at different speeds as well as singular shock solutions for sufficiently large concentrations, for which the mechanism is essentially the same as the singlespecies case. We also consider initial conditions describing a fixed volume of fluid, where solutions are rarefactionshock pairs, and present a comparison to recent experimental results. The longtime behavior of solutions is identified for settled mono and bidisperse suspensions and some leadingorder asymptotics are derived in the singlespecies case for moderate concentrations.
PubDate: 20160516T19:32:17Z
 Abstract: Publication date: Available online 11 May 2016
 Mechanics and polarity in cell motility
 Abstract: Publication date: Available online 12 May 2016
Source:Physica D: Nonlinear Phenomena
Author(s): D. Ambrosi, A. Zanzottera
The motility of a fish keratocyte on a flat substrate exhibits two distinct regimes: the nonmigrating and the migrating one. In both configurations the shape is fixed in time and, when the cell is moving, the velocity is constant in magnitude and direction. Transition from a stable configuration to the other one can be produced by a mechanical or chemotactic perturbation. In order to point out the mechanical nature of such a bistable behaviour, we focus on the actin dynamics inside the cell using a minimal mathematical model. While the protein diffusion, recruitment and segregation govern the polarization process, we show that the free actin mass balance, driven by diffusion, and the polymerized actin retrograde flow, regulated by the active stress, are sufficient ingredients to account for the motile bistability. The length and velocity of the cell are predicted on the basis of the parameters of the substrate and of the cell itself. The key physical ingredient of the theory is the exchange among actin phases at the edges of the cell, that plays a central role both in kinematics and in dynamics.
PubDate: 20160516T19:32:17Z
 Abstract: Publication date: Available online 12 May 2016
 Multibump solutions in a neural field model with external inputs
 Abstract: Publication date: 1 July 2016
Source:Physica D: Nonlinear Phenomena, Volume 326
Author(s): Flora Ferreira, Wolfram Erlhagen, Estela Bicho
We study the conditions for the formation of multiple regions of high activity or “bumps” in a onedimensional, homogeneous neural field with localized inputs. Stable multibump solutions of the integrodifferential equation have been proposed as a model of a neural population representation of remembered external stimuli. We apply a class of oscillatory coupling functions and first derive criteria to the input width and distance, which relate to the synaptic couplings that guarantee the existence and stability of one and two regions of high activity. These inputinduced patterns are attracted by the corresponding stable onebump and twobump solutions when the input is removed. We then extend our analytical and numerical investigation to N bump solutions showing that the constraints on the input shape derived for the twobump case can be exploited to generate a memory of N > 2 localized inputs. We discuss the pattern formation process when either the conditions on the input shape are violated or when the spatial ranges of the excitatory and inhibitory connections are changed. An important aspect for applications is that the theoretical findings allow us to determine for a given coupling function the maximum number of localized inputs that can be stored in a given finite interval.
PubDate: 20160516T19:32:17Z
 Abstract: Publication date: 1 July 2016
 Existence and stability of PTsymmetric states in nonlinear
twodimensional square lattices Abstract: Publication date: 1 July 2016
Source:Physica D: Nonlinear Phenomena, Volume 326
Author(s): Haitao Xu, P.G. Kevrekidis, Dmitry E. Pelinovsky
Solitons and vortices symmetric with respect to simultaneous parity ( P ) and time reversing ( T ) transformations are considered on the square lattice in the framework of the discrete nonlinear Schrödinger equation. The existence and stability of such PT symmetric configurations is analyzed in the limit of weak coupling between the lattice sites, when predictions on the elementary cell of a square lattice (i.e., a single square) can be extended to a large (yet finite) array of lattice cells. In particular, we find all examined vortex configurations are unstable with respect to small perturbations while a branch extending soliton configurations is spectrally stable. Our analytical predictions are found to be in good agreement with numerical computations.
PubDate: 20160516T19:32:17Z
 Abstract: Publication date: 1 July 2016
 Recurrence plots of discretetime Gaussian stochastic processes
 Abstract: Publication date: Available online 9 May 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Sofiane Ramdani, Frédéric Bouchara, Julien Lagarde, Annick Lesne
We investigate the statistical properties of recurrence plots (RPs) of data generated by discretetime stationary Gaussian random processes. We analytically derive the theoretical values of the probabilities of occurrence of recurrence points and consecutive recurrence points forming diagonals on the RP, with an embedding dimension equal to 1 . These results allow us to obtain theoretical values of three measures: (i) the recurrence rate ( R E C ) (ii) the percent determinism ( D E T ) and (iii) RPbased estimation of the ε entropy κ ( ε ) in the sense of correlation entropy. We apply these results to two Gaussian processes, namely the first order autoregressive process and fractional Gaussian noise. For these processes, we simulate a number of realizations and compare the RPbased estimations of the three selected measures to their theoretical values. These comparisons provide useful information on the quality of the estimations, such as the minimum required data length and threshold radius used to construct the RP.
PubDate: 20160510T19:21:51Z
 Abstract: Publication date: Available online 9 May 2016
 Modulational instability and localized breather modes in the discrete
nonlinear Schrödinger equation with helicoidal hopping Abstract: Publication date: Available online 4 May 2016
Source:Physica D: Nonlinear Phenomena
Author(s): J. Stockhofe, P. Schmelcher
We study a onedimensional discrete nonlinear Schrödinger model with hopping to the first and a selected N th neighbor, motivated by a helicoidal arrangement of lattice sites. We provide a detailed analysis of the modulational instability properties of this equation, identifying distinctive multistage instability cascades due to the helicoidal hopping term. Bistability is a characteristic feature of the intrinsically localized breather modes, and it is shown that information on the stability properties of weakly localized solutions can be inferred from the planewave modulational instability results. Based on this argument, we derive analytical estimates of the critical parameters at which the fundamental onsite breather branch of solutions turns unstable. In the limit of large N , these estimates predict the emergence of an effective threshold behavior, which can be viewed as the result of a dimensional crossover to a twodimensional square lattice.
PubDate: 20160505T19:00:55Z
 Abstract: Publication date: Available online 4 May 2016
 Wavelet shrinkage of a noisy dynamical system with nonlinear noise impact
 Abstract: Publication date: 15 June 2016
Source:Physica D: Nonlinear Phenomena, Volume 325
Author(s): Matthieu Garcin, Dominique Guégan
By filtering wavelet coefficients, it is possible to construct a good estimate of a pure signal from noisy data. Especially, for a simple linear noise influence, Donoho and Johnstone (1994) have already defined an optimal filter design in the sense of a minimization of the error made when estimating the pure signal. We set here a different framework where the influence of the noise is nonlinear. In particular, we propose a method to filter the wavelet coefficients of a discrete dynamical system disrupted by a weak noise, in order to construct good estimates of the pure signal, including Bayes’ estimate, minimax estimate, oracular estimate or thresholding estimate. We present the example of a logistic and a Lorenz chaotic dynamical system as well as an adaptation of our technique in order to show empirically the robustness of the thresholding method in presence of leptokurtic noise. Moreover, we test both the hard and the soft thresholding and also another kind of smoother thresholding which seems to have almost the same reconstruction power as the hard thresholding. Finally, besides the tests on an estimated dataset, the method is tested on financial data: oil prices and NOK/USD exchange rate.
PubDate: 20160505T19:00:55Z
 Abstract: Publication date: 15 June 2016
 Firstorder aggregation models with alignment
 Abstract: Publication date: 15 June 2016
Source:Physica D: Nonlinear Phenomena, Volume 325
Author(s): Razvan C. Fetecau, Weiran Sun, Changhui Tan
We include alignment interactions in a wellstudied firstorder attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have nonunique solutions. We investigate the wellposedness of the model and show rigorously how it can be obtained as a macroscopic limit of a secondorder kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.
PubDate: 20160505T19:00:55Z
 Abstract: Publication date: 15 June 2016
 On the evolution of scattering data under perturbations of the Toda
lattice Abstract: Publication date: Available online 25 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): D. Bilman, I. Nenciu
We present the results of an analytical and numerical study of the longtime behavior for certain FermiPastaUlam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doublyinfinite Jacobi matrices, which are wellknown to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: Available online 25 April 2016
 Absolute stability and synchronization in neural field models with
transmission delays Abstract: Publication date: Available online 26 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): ChiuYen Kao, ChihWen Shih, ChangHong Wu
Neural fields model macroscopic parts of the cortex which involve several populations of neurons. We consider a class of neural field models which are represented by integrodifferential equations with transmission time delays which are spacedependent. The considered domains underlying the systems can be bounded or unbounded. A new approach, called sequential contracting, instead of the conventional Lyapunov functional technique, is employed to investigate the global dynamics of such systems. Sufficient conditions for the absolute stability and synchronization of the systems are established. Several numerical examples are presented to demonstrate the theoretical results.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: Available online 26 April 2016
 Correlation functions of the KdV hierarchy and applications to
intersection numbers over M¯g,n Abstract: Publication date: Available online 29 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Marco Bertola, Boris Dubrovin, Di Yang
We derive an explicit generating function of correlations functions of an arbitrary taufunction of the KdV hierarchy. In particular applications, our formulation gives closed formulæ of a new type for the generating series of intersection numbers of ψ classes as well as of mixed ψ  and κ classes in full genera.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: Available online 29 April 2016
 Nonlinear Dynamics on Interconnected Networks
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Alex Arenas, Manlio De Domenico
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 On degree–degree correlations in multilayer networks
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Guilherme Ferraz de Arruda, Emanuele Cozzo, Yamir Moreno, Francisco A. Rodrigues
We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree–degree correlations on multilayer systems, instead of on aggregated networks.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Network bipartivity and the transportation efficiency of European
passenger airlines Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Ernesto Estrada, Jesús GómezGardeñes
The analysis of the structural organization of the interaction network of a complex system is central to understand its functioning. Here, we focus on the analysis of the bipartivity of graphs. We first introduce a mathematical approach to quantify bipartivity and show its implementation in general and random graphs. Then, we tackle the analysis of the transportation networks of European airlines from the point of view of their bipartivity and observe significant differences between traditional and low cost carriers. Bipartivity shows also that alliances and major mergers of traditional airlines provide a way to reduce bipartivity which, in its turn, is closely related to an increase of the transportation efficiency.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Asymptotic periodicity in networks of degradeandfire oscillators
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Alex Blumenthal, Bastien Fernandez
Networks of coupled degradeandfire (DF) oscillators are simple dynamical models of assemblies of interacting selfrepressing genes. For meanfield interactions, which most mathematical studies have assumed so far, every trajectory must approach a periodic orbit. Moreover, asymptotic cluster distributions can be computed explicitly in terms of coupling intensity, and a massive collection of distributions collapses when this intensity passes a threshold. Here, we show that most of these dynamical features persist for an arbitrary coupling topology. In particular, we prove that, in any system of DF oscillators for which in and out coupling weights balance, trajectories with reasonable firing sequences must be asymptotically periodic, and periodic orbits are uniquely determined by their firing sequence. In addition to these structural results, illustrative examples are presented, for which the dynamics can be entirely described.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Cascades in interdependent flow networks
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Antonio Scala, Pier Giorgio De Sanctis Lucentini, Guido Caldarelli, Gregorio D’Agostino
In this manuscript, we investigate the abrupt breakdown behavior of coupled distribution grids under load growth. This scenario mimics the everincreasing customer demand and the foreseen introduction of energy hubs interconnecting the different energy vectors. We extend an analytical model of cascading behavior due to line overloads to the case of interdependent networks and find evidence of first order transitions due to the longrange nature of the flows. Our results indicate that the foreseen increase in the couplings between the grids has two competing effects: on the one hand, it increases the safety region where grids can operate without withstanding systemic failures; on the other hand, it increases the possibility of a joint systems’ failure.
Graphical abstract
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Erosion of synchronization: Coupling heterogeneity and network structure
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
We study the dynamics of networkcoupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with wellknown spectral properties, which allows us to derive further analytical results.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Contactbased model for strategy updating and evolution of cooperation
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Jianlei Zhang, Zengqiang Chen
To establish an available model for the astoundingly strategy decision process of players is not easy, sparking heated debate about the related strategy updating rules is intriguing. Models for evolutionary games have traditionally assumed that players imitate their successful partners by the comparison of respective payoffs, raising the question of what happens if the game information is not easily available. Focusing on this yetunsolved case, the motivation behind the work presented here is to establish a novel model for the updating of states in a spatial population, by detouring the required payoffs in previous models and considering much more players’ contact patterns. It can be handy and understandable to employ switching probabilities for determining the microscopic dynamics of strategy evolution. Our results illuminate the conditions under which the steady coexistence of competing strategies is possible. These findings reveal that the evolutionary fate of the coexisting strategies can be calculated analytically, and provide novel hints for the resolution of cooperative dilemmas in a competitive context. We hope that our results have disclosed new explanations about the survival and coexistence of competing strategies in structured populations.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Consensus dynamics on random rectangular graphs
 Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Ernesto Estrada, Matthew Sheerin
A random rectangular graph (RRG) is a generalization of the random geometric graph (RGG) in which the nodes are embedded into a rectangle with side lengths a and b = 1 / a , instead of on a unit square [ 0 , 1 ] 2 . Two nodes are then connected if and only if they are separated at a Euclidean distance smaller than or equal to a certain threshold radius r . When a = 1 the RRG is identical to the RGG. Here we apply the consensus dynamics model to the RRG. Our main result is a lower bound for the time of consensus, i.e., the time at which the network reaches a global consensus state. To prove this result we need first to find an upper bound for the algebraic connectivity of the RRG, i.e., the second smallest eigenvalue of the combinatorial Laplacian of the graph. This bound is based on a tight lower bound found for the graph diameter. Our results prove that as the rectangle in which the nodes are embedded becomes more elongated, the RRG becomes a ’largeworld’, i.e., the diameter grows to infinity, and a poorlyconnected graph, i.e., the algebraic connectivity decays to zero. The main consequence of these findings is the proof that the time of consensus in RRGs grows to infinity as the rectangle becomes more elongated. In closing, consensus dynamics in RRGs strongly depend on the geometric characteristics of the embedding space, and reaching the consensus state becomes more difficult as the rectangle is more elongated.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Interplay between consensus and coherence in a model of interacting
opinions Abstract: Publication date: 1 June 2016
Source:Physica D: Nonlinear Phenomena, Volumes 323–324
Author(s): Federico Battiston, Andrea Cairoli, Vincenzo Nicosia, Adrian Baule, Vito Latora
The formation of agents’ opinions in a social system is the result of an intricate equilibrium among several driving forces. On the one hand, the social pressure exerted by peers favors the emergence of local consensus. On the other hand, the concurrent participation of agents to discussions on different topics induces each agent to develop a coherent set of opinions across all the topics in which he/she is active. Moreover, the pervasive action of external stimuli, such as mass media, pulls the entire population towards a specific configuration of opinions on different topics. Here we propose a model in which agents with interrelated opinions, interacting on several layers representing different topics, tend to spread their own ideas to their neighborhood, strive to maintain internal coherence, due to the fact that each agent identifies meaningful relationships among its opinions on the different topics, and are at the same time subject to external fields, resembling the pressure of mass media. We show that the presence of heterogeneity in the internal coupling assigned by agents to their different opinions allows to obtain states with mixed levels of consensus, still ensuring that all the agents attain a coherent set of opinions. Furthermore, we show that all the observed features of the model are preserved in the presence of thermal noise up to a critical temperature, after which global consensus is no longer attainable. This suggests the relevance of our results for real social systems, where noise is inevitably present in the form of information uncertainty and misunderstandings. The model also demonstrates how mass media can be effectively used to favor the propagation of a chosen set of opinions, thus polarizing the consensus of an entire population.
PubDate: 20160429T18:42:49Z
 Abstract: Publication date: 1 June 2016
 Quasisteady state reduction for compartmental systems
 Abstract: Publication date: Available online 21 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Alexandra Goeke, Christian Lax
We present a method to determine an asymptotic reduction (in the sense of Tikhonov and Fenichel) for singularly perturbed compartmental systems in the presence of slow transport. It turns out that the reduction can be derived from the individual interaction terms alone. We apply the result to spatially discretized reactiondiffusion systems and obtain (based on the reduced discretized systems) a heuristic to reduce reactiondiffusion systems in presence of slow diffusion.
PubDate: 20160424T18:22:14Z
 Abstract: Publication date: Available online 21 April 2016
 The extended EstabrookWahlquist method
 Abstract: Publication date: Available online 21 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): S. Roy Choudhury, Matthew Russo
Variable Coefficient Korteweg de Vries (vcKdV), modified Korteweg de Vries (vcMKdV), and nonlinear Schröedinger (NLS) equations have a long history dating from their derivation in various applications. A technique based on extended Lax Pairs has been devised recently to derive timeandspacedependentcoefficient generalizations of various such Laxintegrable NLPDE hierarchies, which are thus more general than almost all cases considered earlier via methods such as the Painlevé Test, Bell Polynomials, and similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ‘guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we embark in this paper on an attempt to systematize the derivation of Laxintegrable systems with variable coefficients. We consider the EstabrookWahlquist (EW) prolongation technique, a relatively selfconsistent procedure requiring little prior information. However, this immediately requires that the technique be significantly generalized in several ways, including solving matrix partial differential equations instead of algebraic ones as the structure of the Lax Pair is systematically computed, and also in solving the constraint equations to deduce the explicit forms for various ‘coefficient’ matrices. The new and extended EW technique which results is illustrated by algorithmically deriving generalized Laxintegrable versions of the NLS, generalized fifthorder KdV, MKdV, and derivative nonlinear Schröedinger (DNLS) equations. We also show how this method correctly excludes the existence of a nontrivial Lax pair for a nonintegrable NLPDE such as the variablecoefficient cubicquintic NLS.
PubDate: 20160424T18:22:14Z
 Abstract: Publication date: Available online 21 April 2016
 Cellular replication limits in the LuriaDelbrück mutation model
 Abstract: Publication date: Available online 19 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Ignacio A. RodriguezBrenes, Dominik Wodarz, Natalia L. Komarova
Originally developed to elucidate the mechanisms of natural selection in bacteria, the LuriaDelbrück model assumed that cells are intrinsically capable of dividing an unlimited number of times. This assumption however, is not true for human somatic cells which undergo replicative senescence. Replicative senescence is thought to act as a mechanism to protect against cancer and the escape from it is a ratelimiting step in cancer progression. Here we introduce a LuriaDelbrück model that explicitly takes into account cellular replication limits in the wild type cell population and models the emergence of mutants that escape replicative senescence. We present results on the mean, variance, distribution, and asymptotic behavior of the mutant population in terms of three classical formulations of the problem. More broadly the paper introduces the concept of incorporating replicative limits as part of the LuriaDelbrück mutational framework. Guidelines to extend the theory to include other types of mutations and possible applications to the modeling of telomere crisis and fluctuation analysis are also discussed.
PubDate: 20160419T18:05:31Z
 Abstract: Publication date: Available online 19 April 2016
 A trajectoryfree framework for analysing multiscale systems
 Abstract: Publication date: Available online 19 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Gary Froyland, Georg A. Gottwald, Andy Hammerlindl
We develop algorithms built around properties of the transfer operator and Koopman operator which 1) test for possible multiscale dynamics in a given dynamical system, 2) estimate the magnitude of the timescale separation, and finally 3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem.
PubDate: 20160419T18:05:31Z
 Abstract: Publication date: Available online 19 April 2016
 Multisoliton, multibreather and higher order rogue wave solutions to the
complex short pulse equation Abstract: Publication date: Available online 19 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Liming Ling, BaoFeng Feng, Zuonong Zhu
In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N bright soliton solution in a compact determinant form, the N breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigourously for both the N soliton and the N breather solutions. All three forms of the analytical solutions admit either smoothed, cusped or loopedtype ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.
PubDate: 20160419T18:05:31Z
 Abstract: Publication date: Available online 19 April 2016
 Nonlinear optical vibrations of singlewalled carbon nanotubes. 1. Energy
exchange and localization of lowfrequency oscillations Abstract: Publication date: Available online 4 April 2016
Source:Physica D: Nonlinear Phenomena
Author(s): V.V. Smirnov, L.I. Manevitch, M. Strozzi, F. Pellicano
We present the results of analytical study and molecular dynamics simulation of low energy nonlinear nonstationary dynamics of singlewalled carbon nanotubes (CNTs). New phenomena of intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT are analytically predicted in the framework of the continuum shell theory. Their origin is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the molecular dynamics simulation of simply supported CNTs.
PubDate: 20160404T17:26:25Z
 Abstract: Publication date: Available online 4 April 2016
 Inertial effects on thinfilm wave structures with imposed surface shear
on an inclined plane Abstract: Publication date: Available online 21 March 2016
Source:Physica D: Nonlinear Phenomena
Author(s): M. Sivapuratharasu, S. Hibberd, M.E. Hubbard, H. Power
This study provides an extended approach to the mathematical simulation of thinfilm flow on a flat inclined plane relevant to flows subject to high surface shear. Motivated by modelling thinfilm structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio ε . Approximations are made assuming a Reynolds number, Re ∼ O ( ε − 1 ) and depthaveraging used to simplify the governing NavierStokes equations. A parallel Stokes flow is expected in the absence of any wave disturbance and a generalisation for the flow is based on a local quadratic profile. This approch provides a more general system which includes inertial effects and is solved numerically. Flow structures are compared with studies for Stokes flow in the limit of negligible inertial effects. Both twotier and threetier wave disturbances are used to study film profile evolution. A parametric study is provided for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken and identifies new profiles not previously predicted when inertial effects are neglected.
PubDate: 20160325T13:22:38Z
 Abstract: Publication date: Available online 21 March 2016
 Modulational instability in nonlinear nonlocal equations of regularized
long wave type Abstract: Publication date: Available online 18 March 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Vera Mikyoung Hur, Ashish Kumar Pandey
We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of BenjaminBonaMahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Kortewegde Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently small, periodic traveling wave of the regularized Boussinesq equation is stable to square integrable perturbations.
PubDate: 20160321T13:19:25Z
 Abstract: Publication date: Available online 18 March 2016
 Jump bifurcations in some degenerate planar piecewise linear differential
systems with three zones Abstract: Publication date: Available online 11 March 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Rodrigo Euzébio, Rubens Pazim, Enrique Ponce
We consider continuous piecewiselinear differential systems with three zones where the central one is degenerate, that is, the determinant of its linear part vanishes. By moving one parameter which is associated to the equilibrium position, we detect some new bifurcations exhibiting jump transitions both in the equilibrium location and in the appearance of limit cycles. In particular, we introduce the scabbard bifurcation, characterized by the birth of a limit cycle from a continuum of equilibrium points. Some of the studied bifurcations are detected, after an appropriate choice of parameters, in a piecewise linear MorrisLecar model for the activity of a single neuron activity, which is usually considered as a reduction of the celebrated HodgkinHuxley equations.
PubDate: 20160316T13:03:26Z
 Abstract: Publication date: Available online 11 March 2016
 A numerical method for computing initial conditions of Lagrangian
invariant tori using the frequency map Abstract: Publication date: Available online 10 March 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Alejandro Luque, Jordi Villanueva
We present a numerical method for computing initial conditions of Lagrangian quasiperiodic invariant tori of Hamiltonian systems and symplectic maps. Such initial conditions are found by solving, using the Newton method, a nonlinear system obtained by imposing suitable conditions on the frequency map. The basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasiperiodic signal, allowing to compute frequencies and their derivatives with respect to parameters. Roughly speaking, this method consists in computing suitable weighted averages of the iterates of the signal and using the Richardson extrapolation method. The proposed approach performs with high accuracy at a moderate computational cost. We illustrate the method by considering a discrete FPU model and the vicinity of the point L 4 in a RTBP.
PubDate: 20160311T12:45:40Z
 Abstract: Publication date: Available online 10 March 2016
 Dressing method for the vector sineGordon equation and its soliton
interactions Abstract: Publication date: Available online 9 March 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Alexander V. Mikhailov, Georgios Papamikos, Jing Ping Wang
In this paper, we develop the dressing method to study the exact solutions for the vector sineGordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multisoliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices.
PubDate: 20160311T12:45:40Z
 Abstract: Publication date: Available online 9 March 2016
 Three–dimensional representations of the tube manifolds of the
planar restricted three–body problem Abstract: Publication date: Available online 10 March 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Elena Lega, Massimiliano Guzzo
The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium points L1, L2 play a key role in the understanding of the complicated dynamics of the circular restricted three–body problem. By developing a recent technique of computation of the stable and unstable manifolds, based on the use of Fast Lyapunov Indicators modified by the introduction of a filtering window function, we compute sample three–dimensional representations of the manifolds which show an original vista about their complicated development in the phasespace.
PubDate: 20160311T12:45:40Z
 Abstract: Publication date: Available online 10 March 2016
 Hopf normal form with SN symmetry and reduction to systems of nonlinearly
coupled phase oscillators Abstract: Publication date: Available online 26 February 2016
Source:Physica D: Nonlinear Phenomena
Author(s): Peter Ashwin, Ana Rodrigues
Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry S N are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive  and we characterise generic multiway interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, ϵ (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small ϵ > 0 . Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with S N symmetry. For fixed N and taking the limit 0 < ϵ ≪ λ ≪ 1 , we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the wellknown KuramotoSakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of N . Using a normalization that maintains nontrivial interactions in the limit N → ∞ , we show that the additional terms can lead to new phenomena in terms of coexistence of twocluster states with the same phase difference but different cluster size.
PubDate: 20160307T12:36:02Z
 Abstract: Publication date: Available online 26 February 2016
 Traveling wave solutions in a chain of periodically forced coupled
nonlinear oscillators Abstract: Publication date: Available online 27 February 2016
Source:Physica D: Nonlinear Phenomena
Author(s): M. Duanmu, N. Whitaker, P.G. Kevrekidis, A. Vainchtein, J.E. Rubin
Motivated by earlier studies of artificial perceptions of light called phosphenes, we analyze traveling wave solutions in a chain of periodically forced coupled nonlinear oscillators modeling this phenomenon. We examine the discrete model problem in its cotraveling frame and systematically obtain the corresponding traveling waves in one spatial dimension. Direct numerical simulations as well as linear stability analysis are employed to reveal the parameter regions where the traveling waves are stable, and these waves are, in turn, connected to the standing waves analyzed in earlier work. We also consider a twodimensional extension of the model and demonstrate the robust evolution and stability of planar fronts. Our simulations also suggest the radial fronts tend to either annihilate or expand and flatten out, depending on the phase value inside and the parameter regime. Finally, we observe that solutions that initially feature two symmetric fronts with bulged centers evolve in qualitative agreement with experimental observations of phosphenes.
PubDate: 20160307T12:36:02Z
 Abstract: Publication date: Available online 27 February 2016
 Filter accuracy for the Lorenz 96 model: Fixed versus adaptive observation
operators Abstract: Publication date: Available online 23 February 2016
Source:Physica D: Nonlinear Phenomena
Author(s): K.J.H. Law, D. SanzAlonso, A. Shukla, A.M. Stuart
In the context of filtering chaotic dynamical systems it is wellknown that partial observations, if sufficiently informative, can be used to control the inherent uncertainty due to chaos. The purpose of this paper is to investigate, both theoretically and numerically, conditions on the observations of chaotic systems under which they can be accurately filtered. In particular, we highlight the advantage of adaptive observation operators over fixed ones. The Lorenz ’96 model is used to exemplify our findings. We consider discretetime and continuoustime observations in our theoretical developments. We prove that, for fixed observation operator, the 3DVAR filter can recover the system state within a neighbourhood determined by the size of the observational noise. It is required that a sufficiently large proportion of the state vector is observed, and an explicit form for such sufficient fixed observation operator is given. Numerical experiments, where the data is incorporated by use of the 3DVAR and extended Kalman filters, suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction. Adaptive observation operators are then studied numerically; we show that, for carefully chosen adaptive observation operators, the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator. Indeed, we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive Lyapunov exponents of the underlying system.
PubDate: 20160224T12:04:11Z
 Abstract: Publication date: Available online 23 February 2016